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Copyright 2026 The Formal Conjectures Authors.
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import FormalConjecturesUtil
open Filter Realopen scoped ArithmeticFunction.omega
namespace Erdos1203
If $\omega(n)$ counts the number of distinct prime divisors of $n$ then let $F(n)=\max_k \omega(n+k)\frac{\log\log k}{\log k}.$
noncomputable def F (n : ℕ) : ℝ :=
⨆ k : ℕ, (ω (n + k) : ℝ) * (log (log (k : ℝ)) / log (k : ℝ))
Prove that $F(n)\to \infty$ as $n\to \infty$.
@[category research open, AMS 11]
theorem erdos_1203 :
answer(sorry) ↔ Tendsto F atTop atTop := ⊢ True ↔ Tendsto F atTop atTop
All goals completed! 🐙
It is easy to prove that $F(n)\geq 1-o(1)$.
@[category research solved, AMS 11]
theorem erdos_1203.variants.lower_bound :
∀ ε > 0, ∀ᶠ n in atTop, F n ≥ 1 - ε := ⊢ ∀ ε > 0, ∀ᶠ (n : ℕ) in atTop, F n ≥ 1 - ε
All goals completed! 🐙
end Erdos1203